A man who went out between 3 and 4 and returned between 8 and 9, found that the hands of the watch had exactly changed places. He returned at:

As per the given above question , we can say that
1. In 1 min ,minute hand moves 6 Degree
2. In 1 min,hour hand moves 0.5 Degree
Let suppose man went out at y past three i.e. 3 hours y minutes
then, minute hand will be at ( 6y ) Degree from origin.
and hour hand will be at ( 3 × 30 + y × 0.5) Degree = ( 90 + 0.5y ) Degree
Similarly ,man returned at z past eight i.e. 8 hours z minutes
then, minute hand will be at ( 6z ) Degree from origin.
and hour hand will be at ( 8 × 30 + z × 0.5 ) Degree = ( 240 + 0.5z ) Degree
And as per the given condition :-
6y = 90 + 0.5y ...... ( ⅰ )
and 6z = 240 + 0.5z ...... ( ⅱ )
Solve these equations , we get y = 16.36 and z =43.63